Properties

Label 475.2.b.c.324.3
Level $475$
Weight $2$
Character 475.324
Analytic conductor $3.793$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(324,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.3
Root \(-0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 475.324
Dual form 475.2.b.c.324.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.246980i q^{2} +0.801938i q^{3} +1.93900 q^{4} +0.198062 q^{6} +1.69202i q^{7} -0.972853i q^{8} +2.35690 q^{9} +O(q^{10})\) \(q-0.246980i q^{2} +0.801938i q^{3} +1.93900 q^{4} +0.198062 q^{6} +1.69202i q^{7} -0.972853i q^{8} +2.35690 q^{9} -0.911854 q^{11} +1.55496i q^{12} -1.55496i q^{13} +0.417895 q^{14} +3.63773 q^{16} +5.29590i q^{17} -0.582105i q^{18} +1.00000 q^{19} -1.35690 q^{21} +0.225209i q^{22} -4.24698i q^{23} +0.780167 q^{24} -0.384043 q^{26} +4.29590i q^{27} +3.28083i q^{28} -5.00969 q^{29} +1.82908 q^{31} -2.84415i q^{32} -0.731250i q^{33} +1.30798 q^{34} +4.57002 q^{36} +6.29590i q^{37} -0.246980i q^{38} +1.24698 q^{39} +4.18060 q^{41} +0.335126i q^{42} -7.31767i q^{43} -1.76809 q^{44} -1.04892 q^{46} -2.04892i q^{47} +2.91723i q^{48} +4.13706 q^{49} -4.24698 q^{51} -3.01507i q^{52} +2.70171i q^{53} +1.06100 q^{54} +1.64609 q^{56} +0.801938i q^{57} +1.23729i q^{58} -9.87800 q^{59} +0.542877 q^{61} -0.451747i q^{62} +3.98792i q^{63} +6.57301 q^{64} -0.180604 q^{66} -13.9976i q^{67} +10.2687i q^{68} +3.40581 q^{69} -12.8780 q^{71} -2.29291i q^{72} +2.80731i q^{73} +1.55496 q^{74} +1.93900 q^{76} -1.54288i q^{77} -0.307979i q^{78} -1.59419 q^{79} +3.62565 q^{81} -1.03252i q^{82} -12.2349i q^{83} -2.63102 q^{84} -1.80731 q^{86} -4.01746i q^{87} +0.887100i q^{88} -2.91723 q^{89} +2.63102 q^{91} -8.23490i q^{92} +1.46681i q^{93} -0.506041 q^{94} +2.28083 q^{96} +1.55496i q^{97} -1.02177i q^{98} -2.14914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9} + 2 q^{11} + 14 q^{14} + 36 q^{16} + 6 q^{19} + 2 q^{24} + 18 q^{26} + 14 q^{29} - 10 q^{31} + 18 q^{34} - 22 q^{36} - 2 q^{39} + 2 q^{41} + 30 q^{44} + 12 q^{46} + 14 q^{49} - 16 q^{51} + 26 q^{54} - 70 q^{56} - 20 q^{59} - 34 q^{61} - 98 q^{64} + 22 q^{66} - 6 q^{69} - 38 q^{71} + 10 q^{74} - 8 q^{76} - 36 q^{79} - 2 q^{81} + 14 q^{84} + 4 q^{86} - 4 q^{89} - 14 q^{91} - 22 q^{94} + 36 q^{96} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.246980i − 0.174641i −0.996180 0.0873205i \(-0.972170\pi\)
0.996180 0.0873205i \(-0.0278304\pi\)
\(3\) 0.801938i 0.462999i 0.972835 + 0.231499i \(0.0743632\pi\)
−0.972835 + 0.231499i \(0.925637\pi\)
\(4\) 1.93900 0.969501
\(5\) 0 0
\(6\) 0.198062 0.0808586
\(7\) 1.69202i 0.639524i 0.947498 + 0.319762i \(0.103603\pi\)
−0.947498 + 0.319762i \(0.896397\pi\)
\(8\) − 0.972853i − 0.343955i
\(9\) 2.35690 0.785632
\(10\) 0 0
\(11\) −0.911854 −0.274934 −0.137467 0.990506i \(-0.543896\pi\)
−0.137467 + 0.990506i \(0.543896\pi\)
\(12\) 1.55496i 0.448878i
\(13\) − 1.55496i − 0.431268i −0.976474 0.215634i \(-0.930818\pi\)
0.976474 0.215634i \(-0.0691818\pi\)
\(14\) 0.417895 0.111687
\(15\) 0 0
\(16\) 3.63773 0.909432
\(17\) 5.29590i 1.28444i 0.766519 + 0.642222i \(0.221987\pi\)
−0.766519 + 0.642222i \(0.778013\pi\)
\(18\) − 0.582105i − 0.137204i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.35690 −0.296099
\(22\) 0.225209i 0.0480148i
\(23\) − 4.24698i − 0.885556i −0.896631 0.442778i \(-0.853993\pi\)
0.896631 0.442778i \(-0.146007\pi\)
\(24\) 0.780167 0.159251
\(25\) 0 0
\(26\) −0.384043 −0.0753170
\(27\) 4.29590i 0.826746i
\(28\) 3.28083i 0.620019i
\(29\) −5.00969 −0.930276 −0.465138 0.885238i \(-0.653995\pi\)
−0.465138 + 0.885238i \(0.653995\pi\)
\(30\) 0 0
\(31\) 1.82908 0.328513 0.164257 0.986418i \(-0.447477\pi\)
0.164257 + 0.986418i \(0.447477\pi\)
\(32\) − 2.84415i − 0.502779i
\(33\) − 0.731250i − 0.127294i
\(34\) 1.30798 0.224316
\(35\) 0 0
\(36\) 4.57002 0.761671
\(37\) 6.29590i 1.03504i 0.855671 + 0.517520i \(0.173145\pi\)
−0.855671 + 0.517520i \(0.826855\pi\)
\(38\) − 0.246980i − 0.0400654i
\(39\) 1.24698 0.199677
\(40\) 0 0
\(41\) 4.18060 0.652901 0.326450 0.945214i \(-0.394147\pi\)
0.326450 + 0.945214i \(0.394147\pi\)
\(42\) 0.335126i 0.0517110i
\(43\) − 7.31767i − 1.11593i −0.829863 0.557967i \(-0.811582\pi\)
0.829863 0.557967i \(-0.188418\pi\)
\(44\) −1.76809 −0.266549
\(45\) 0 0
\(46\) −1.04892 −0.154654
\(47\) − 2.04892i − 0.298865i −0.988772 0.149433i \(-0.952255\pi\)
0.988772 0.149433i \(-0.0477447\pi\)
\(48\) 2.91723i 0.421066i
\(49\) 4.13706 0.591009
\(50\) 0 0
\(51\) −4.24698 −0.594696
\(52\) − 3.01507i − 0.418114i
\(53\) 2.70171i 0.371108i 0.982634 + 0.185554i \(0.0594080\pi\)
−0.982634 + 0.185554i \(0.940592\pi\)
\(54\) 1.06100 0.144384
\(55\) 0 0
\(56\) 1.64609 0.219968
\(57\) 0.801938i 0.106219i
\(58\) 1.23729i 0.162464i
\(59\) −9.87800 −1.28601 −0.643003 0.765864i \(-0.722312\pi\)
−0.643003 + 0.765864i \(0.722312\pi\)
\(60\) 0 0
\(61\) 0.542877 0.0695082 0.0347541 0.999396i \(-0.488935\pi\)
0.0347541 + 0.999396i \(0.488935\pi\)
\(62\) − 0.451747i − 0.0573719i
\(63\) 3.98792i 0.502430i
\(64\) 6.57301 0.821626
\(65\) 0 0
\(66\) −0.180604 −0.0222308
\(67\) − 13.9976i − 1.71008i −0.518562 0.855040i \(-0.673533\pi\)
0.518562 0.855040i \(-0.326467\pi\)
\(68\) 10.2687i 1.24527i
\(69\) 3.40581 0.410012
\(70\) 0 0
\(71\) −12.8780 −1.52834 −0.764169 0.645016i \(-0.776851\pi\)
−0.764169 + 0.645016i \(0.776851\pi\)
\(72\) − 2.29291i − 0.270222i
\(73\) 2.80731i 0.328571i 0.986413 + 0.164286i \(0.0525319\pi\)
−0.986413 + 0.164286i \(0.947468\pi\)
\(74\) 1.55496 0.180760
\(75\) 0 0
\(76\) 1.93900 0.222419
\(77\) − 1.54288i − 0.175827i
\(78\) − 0.307979i − 0.0348717i
\(79\) −1.59419 −0.179360 −0.0896800 0.995971i \(-0.528584\pi\)
−0.0896800 + 0.995971i \(0.528584\pi\)
\(80\) 0 0
\(81\) 3.62565 0.402850
\(82\) − 1.03252i − 0.114023i
\(83\) − 12.2349i − 1.34295i −0.741025 0.671477i \(-0.765660\pi\)
0.741025 0.671477i \(-0.234340\pi\)
\(84\) −2.63102 −0.287068
\(85\) 0 0
\(86\) −1.80731 −0.194888
\(87\) − 4.01746i − 0.430717i
\(88\) 0.887100i 0.0945652i
\(89\) −2.91723 −0.309226 −0.154613 0.987975i \(-0.549413\pi\)
−0.154613 + 0.987975i \(0.549413\pi\)
\(90\) 0 0
\(91\) 2.63102 0.275806
\(92\) − 8.23490i − 0.858547i
\(93\) 1.46681i 0.152101i
\(94\) −0.506041 −0.0521941
\(95\) 0 0
\(96\) 2.28083 0.232786
\(97\) 1.55496i 0.157882i 0.996879 + 0.0789410i \(0.0251539\pi\)
−0.996879 + 0.0789410i \(0.974846\pi\)
\(98\) − 1.02177i − 0.103214i
\(99\) −2.14914 −0.215997
\(100\) 0 0
\(101\) −16.6015 −1.65191 −0.825955 0.563737i \(-0.809363\pi\)
−0.825955 + 0.563737i \(0.809363\pi\)
\(102\) 1.04892i 0.103858i
\(103\) 4.84548i 0.477439i 0.971089 + 0.238720i \(0.0767277\pi\)
−0.971089 + 0.238720i \(0.923272\pi\)
\(104\) −1.51275 −0.148337
\(105\) 0 0
\(106\) 0.667267 0.0648107
\(107\) − 4.46681i − 0.431823i −0.976413 0.215912i \(-0.930728\pi\)
0.976413 0.215912i \(-0.0692723\pi\)
\(108\) 8.32975i 0.801530i
\(109\) −18.8267 −1.80327 −0.901635 0.432498i \(-0.857632\pi\)
−0.901635 + 0.432498i \(0.857632\pi\)
\(110\) 0 0
\(111\) −5.04892 −0.479222
\(112\) 6.15511i 0.581603i
\(113\) − 20.0368i − 1.88491i −0.334337 0.942453i \(-0.608512\pi\)
0.334337 0.942453i \(-0.391488\pi\)
\(114\) 0.198062 0.0185502
\(115\) 0 0
\(116\) −9.71379 −0.901903
\(117\) − 3.66487i − 0.338818i
\(118\) 2.43967i 0.224589i
\(119\) −8.96077 −0.821433
\(120\) 0 0
\(121\) −10.1685 −0.924411
\(122\) − 0.134079i − 0.0121390i
\(123\) 3.35258i 0.302292i
\(124\) 3.54660 0.318494
\(125\) 0 0
\(126\) 0.984935 0.0877449
\(127\) − 17.8702i − 1.58573i −0.609399 0.792863i \(-0.708589\pi\)
0.609399 0.792863i \(-0.291411\pi\)
\(128\) − 7.31170i − 0.646269i
\(129\) 5.86831 0.516676
\(130\) 0 0
\(131\) 7.44265 0.650267 0.325134 0.945668i \(-0.394591\pi\)
0.325134 + 0.945668i \(0.394591\pi\)
\(132\) − 1.41789i − 0.123412i
\(133\) 1.69202i 0.146717i
\(134\) −3.45712 −0.298650
\(135\) 0 0
\(136\) 5.15213 0.441791
\(137\) 5.68664i 0.485843i 0.970046 + 0.242921i \(0.0781057\pi\)
−0.970046 + 0.242921i \(0.921894\pi\)
\(138\) − 0.841166i − 0.0716048i
\(139\) −3.61596 −0.306701 −0.153351 0.988172i \(-0.549006\pi\)
−0.153351 + 0.988172i \(0.549006\pi\)
\(140\) 0 0
\(141\) 1.64310 0.138374
\(142\) 3.18060i 0.266910i
\(143\) 1.41789i 0.118570i
\(144\) 8.57374 0.714479
\(145\) 0 0
\(146\) 0.693349 0.0573820
\(147\) 3.31767i 0.273637i
\(148\) 12.2078i 1.00347i
\(149\) −3.29052 −0.269570 −0.134785 0.990875i \(-0.543034\pi\)
−0.134785 + 0.990875i \(0.543034\pi\)
\(150\) 0 0
\(151\) −10.2131 −0.831133 −0.415566 0.909563i \(-0.636417\pi\)
−0.415566 + 0.909563i \(0.636417\pi\)
\(152\) − 0.972853i − 0.0789088i
\(153\) 12.4819i 1.00910i
\(154\) −0.381059 −0.0307066
\(155\) 0 0
\(156\) 2.41789 0.193587
\(157\) 14.3448i 1.14484i 0.819960 + 0.572420i \(0.193995\pi\)
−0.819960 + 0.572420i \(0.806005\pi\)
\(158\) 0.393732i 0.0313236i
\(159\) −2.16660 −0.171823
\(160\) 0 0
\(161\) 7.18598 0.566335
\(162\) − 0.895461i − 0.0703540i
\(163\) 19.5308i 1.52977i 0.644167 + 0.764885i \(0.277204\pi\)
−0.644167 + 0.764885i \(0.722796\pi\)
\(164\) 8.10620 0.632988
\(165\) 0 0
\(166\) −3.02177 −0.234535
\(167\) 11.8823i 0.919481i 0.888053 + 0.459741i \(0.152058\pi\)
−0.888053 + 0.459741i \(0.847942\pi\)
\(168\) 1.32006i 0.101845i
\(169\) 10.5821 0.814008
\(170\) 0 0
\(171\) 2.35690 0.180236
\(172\) − 14.1890i − 1.08190i
\(173\) − 15.4722i − 1.17633i −0.808741 0.588164i \(-0.799851\pi\)
0.808741 0.588164i \(-0.200149\pi\)
\(174\) −0.992230 −0.0752208
\(175\) 0 0
\(176\) −3.31708 −0.250034
\(177\) − 7.92154i − 0.595420i
\(178\) 0.720497i 0.0540035i
\(179\) −2.16421 −0.161761 −0.0808803 0.996724i \(-0.525773\pi\)
−0.0808803 + 0.996724i \(0.525773\pi\)
\(180\) 0 0
\(181\) 16.8974 1.25597 0.627986 0.778225i \(-0.283879\pi\)
0.627986 + 0.778225i \(0.283879\pi\)
\(182\) − 0.649809i − 0.0481670i
\(183\) 0.435353i 0.0321822i
\(184\) −4.13169 −0.304592
\(185\) 0 0
\(186\) 0.362273 0.0265631
\(187\) − 4.82908i − 0.353138i
\(188\) − 3.97285i − 0.289750i
\(189\) −7.26875 −0.528724
\(190\) 0 0
\(191\) 5.92394 0.428641 0.214320 0.976763i \(-0.431246\pi\)
0.214320 + 0.976763i \(0.431246\pi\)
\(192\) 5.27114i 0.380412i
\(193\) 4.43535i 0.319264i 0.987177 + 0.159632i \(0.0510307\pi\)
−0.987177 + 0.159632i \(0.948969\pi\)
\(194\) 0.384043 0.0275727
\(195\) 0 0
\(196\) 8.02177 0.572984
\(197\) 16.4722i 1.17359i 0.809734 + 0.586797i \(0.199612\pi\)
−0.809734 + 0.586797i \(0.800388\pi\)
\(198\) 0.530795i 0.0377220i
\(199\) 24.4131 1.73060 0.865300 0.501255i \(-0.167128\pi\)
0.865300 + 0.501255i \(0.167128\pi\)
\(200\) 0 0
\(201\) 11.2252 0.791765
\(202\) 4.10023i 0.288491i
\(203\) − 8.47650i − 0.594934i
\(204\) −8.23490 −0.576558
\(205\) 0 0
\(206\) 1.19673 0.0833804
\(207\) − 10.0097i − 0.695721i
\(208\) − 5.65651i − 0.392209i
\(209\) −0.911854 −0.0630743
\(210\) 0 0
\(211\) −4.34050 −0.298813 −0.149406 0.988776i \(-0.547736\pi\)
−0.149406 + 0.988776i \(0.547736\pi\)
\(212\) 5.23862i 0.359790i
\(213\) − 10.3274i − 0.707619i
\(214\) −1.10321 −0.0754140
\(215\) 0 0
\(216\) 4.17928 0.284364
\(217\) 3.09485i 0.210092i
\(218\) 4.64981i 0.314925i
\(219\) −2.25129 −0.152128
\(220\) 0 0
\(221\) 8.23490 0.553939
\(222\) 1.24698i 0.0836918i
\(223\) − 26.2379i − 1.75702i −0.477725 0.878509i \(-0.658539\pi\)
0.477725 0.878509i \(-0.341461\pi\)
\(224\) 4.81236 0.321540
\(225\) 0 0
\(226\) −4.94869 −0.329182
\(227\) 14.6853i 0.974699i 0.873207 + 0.487349i \(0.162036\pi\)
−0.873207 + 0.487349i \(0.837964\pi\)
\(228\) 1.55496i 0.102980i
\(229\) 21.6407 1.43006 0.715029 0.699095i \(-0.246413\pi\)
0.715029 + 0.699095i \(0.246413\pi\)
\(230\) 0 0
\(231\) 1.23729 0.0814078
\(232\) 4.87369i 0.319973i
\(233\) − 27.1183i − 1.77658i −0.459286 0.888289i \(-0.651895\pi\)
0.459286 0.888289i \(-0.348105\pi\)
\(234\) −0.905149 −0.0591715
\(235\) 0 0
\(236\) −19.1535 −1.24678
\(237\) − 1.27844i − 0.0830435i
\(238\) 2.21313i 0.143456i
\(239\) 11.5308 0.745865 0.372933 0.927858i \(-0.378352\pi\)
0.372933 + 0.927858i \(0.378352\pi\)
\(240\) 0 0
\(241\) 11.8194 0.761354 0.380677 0.924708i \(-0.375691\pi\)
0.380677 + 0.924708i \(0.375691\pi\)
\(242\) 2.51142i 0.161440i
\(243\) 15.7952i 1.01326i
\(244\) 1.05264 0.0673883
\(245\) 0 0
\(246\) 0.828020 0.0527926
\(247\) − 1.55496i − 0.0989396i
\(248\) − 1.77943i − 0.112994i
\(249\) 9.81163 0.621787
\(250\) 0 0
\(251\) −9.66487 −0.610041 −0.305021 0.952346i \(-0.598663\pi\)
−0.305021 + 0.952346i \(0.598663\pi\)
\(252\) 7.73258i 0.487107i
\(253\) 3.87263i 0.243470i
\(254\) −4.41358 −0.276933
\(255\) 0 0
\(256\) 11.3402 0.708761
\(257\) 14.1860i 0.884897i 0.896794 + 0.442449i \(0.145890\pi\)
−0.896794 + 0.442449i \(0.854110\pi\)
\(258\) − 1.44935i − 0.0902328i
\(259\) −10.6528 −0.661932
\(260\) 0 0
\(261\) −11.8073 −0.730854
\(262\) − 1.83818i − 0.113563i
\(263\) 21.7942i 1.34389i 0.740603 + 0.671943i \(0.234540\pi\)
−0.740603 + 0.671943i \(0.765460\pi\)
\(264\) −0.711399 −0.0437836
\(265\) 0 0
\(266\) 0.417895 0.0256228
\(267\) − 2.33944i − 0.143171i
\(268\) − 27.1414i − 1.65792i
\(269\) 24.7265 1.50760 0.753800 0.657104i \(-0.228219\pi\)
0.753800 + 0.657104i \(0.228219\pi\)
\(270\) 0 0
\(271\) −13.2295 −0.803636 −0.401818 0.915720i \(-0.631622\pi\)
−0.401818 + 0.915720i \(0.631622\pi\)
\(272\) 19.2650i 1.16811i
\(273\) 2.10992i 0.127698i
\(274\) 1.40449 0.0848481
\(275\) 0 0
\(276\) 6.60388 0.397507
\(277\) − 0.560335i − 0.0336673i −0.999858 0.0168336i \(-0.994641\pi\)
0.999858 0.0168336i \(-0.00535856\pi\)
\(278\) 0.893068i 0.0535626i
\(279\) 4.31096 0.258091
\(280\) 0 0
\(281\) 27.6039 1.64671 0.823355 0.567527i \(-0.192100\pi\)
0.823355 + 0.567527i \(0.192100\pi\)
\(282\) − 0.405813i − 0.0241658i
\(283\) 15.9608i 0.948769i 0.880318 + 0.474385i \(0.157329\pi\)
−0.880318 + 0.474385i \(0.842671\pi\)
\(284\) −24.9705 −1.48172
\(285\) 0 0
\(286\) 0.350191 0.0207072
\(287\) 7.07367i 0.417546i
\(288\) − 6.70337i − 0.395000i
\(289\) −11.0465 −0.649796
\(290\) 0 0
\(291\) −1.24698 −0.0730992
\(292\) 5.44339i 0.318550i
\(293\) − 25.3327i − 1.47995i −0.672632 0.739977i \(-0.734836\pi\)
0.672632 0.739977i \(-0.265164\pi\)
\(294\) 0.819396 0.0477882
\(295\) 0 0
\(296\) 6.12498 0.356007
\(297\) − 3.91723i − 0.227301i
\(298\) 0.812691i 0.0470779i
\(299\) −6.60388 −0.381912
\(300\) 0 0
\(301\) 12.3817 0.713666
\(302\) 2.52243i 0.145150i
\(303\) − 13.3134i − 0.764832i
\(304\) 3.63773 0.208638
\(305\) 0 0
\(306\) 3.08277 0.176230
\(307\) − 11.5574i − 0.659613i −0.944049 0.329806i \(-0.893017\pi\)
0.944049 0.329806i \(-0.106983\pi\)
\(308\) − 2.99164i − 0.170464i
\(309\) −3.88577 −0.221054
\(310\) 0 0
\(311\) −18.3424 −1.04010 −0.520052 0.854135i \(-0.674087\pi\)
−0.520052 + 0.854135i \(0.674087\pi\)
\(312\) − 1.21313i − 0.0686798i
\(313\) 18.9119i 1.06896i 0.845181 + 0.534481i \(0.179493\pi\)
−0.845181 + 0.534481i \(0.820507\pi\)
\(314\) 3.54288 0.199936
\(315\) 0 0
\(316\) −3.09113 −0.173890
\(317\) 20.2784i 1.13895i 0.822008 + 0.569475i \(0.192854\pi\)
−0.822008 + 0.569475i \(0.807146\pi\)
\(318\) 0.535107i 0.0300073i
\(319\) 4.56810 0.255765
\(320\) 0 0
\(321\) 3.58211 0.199934
\(322\) − 1.77479i − 0.0989052i
\(323\) 5.29590i 0.294672i
\(324\) 7.03013 0.390563
\(325\) 0 0
\(326\) 4.82371 0.267160
\(327\) − 15.0978i − 0.834912i
\(328\) − 4.06711i − 0.224569i
\(329\) 3.46681 0.191132
\(330\) 0 0
\(331\) −4.77479 −0.262446 −0.131223 0.991353i \(-0.541890\pi\)
−0.131223 + 0.991353i \(0.541890\pi\)
\(332\) − 23.7235i − 1.30200i
\(333\) 14.8388i 0.813160i
\(334\) 2.93469 0.160579
\(335\) 0 0
\(336\) −4.93602 −0.269282
\(337\) 24.3967i 1.32897i 0.747300 + 0.664487i \(0.231350\pi\)
−0.747300 + 0.664487i \(0.768650\pi\)
\(338\) − 2.61356i − 0.142159i
\(339\) 16.0683 0.872710
\(340\) 0 0
\(341\) −1.66786 −0.0903196
\(342\) − 0.582105i − 0.0314766i
\(343\) 18.8442i 1.01749i
\(344\) −7.11901 −0.383832
\(345\) 0 0
\(346\) −3.82132 −0.205435
\(347\) − 9.99761i − 0.536700i −0.963322 0.268350i \(-0.913522\pi\)
0.963322 0.268350i \(-0.0864783\pi\)
\(348\) − 7.78986i − 0.417580i
\(349\) −21.9584 −1.17541 −0.587703 0.809077i \(-0.699967\pi\)
−0.587703 + 0.809077i \(0.699967\pi\)
\(350\) 0 0
\(351\) 6.67994 0.356549
\(352\) 2.59345i 0.138231i
\(353\) 36.8786i 1.96285i 0.191848 + 0.981425i \(0.438552\pi\)
−0.191848 + 0.981425i \(0.561448\pi\)
\(354\) −1.95646 −0.103985
\(355\) 0 0
\(356\) −5.65651 −0.299795
\(357\) − 7.18598i − 0.380322i
\(358\) 0.534516i 0.0282500i
\(359\) −16.5187 −0.871824 −0.435912 0.899989i \(-0.643574\pi\)
−0.435912 + 0.899989i \(0.643574\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 4.17331i − 0.219344i
\(363\) − 8.15452i − 0.428001i
\(364\) 5.10156 0.267394
\(365\) 0 0
\(366\) 0.107523 0.00562034
\(367\) − 6.83148i − 0.356600i −0.983976 0.178300i \(-0.942940\pi\)
0.983976 0.178300i \(-0.0570598\pi\)
\(368\) − 15.4494i − 0.805353i
\(369\) 9.85325 0.512940
\(370\) 0 0
\(371\) −4.57135 −0.237333
\(372\) 2.84415i 0.147462i
\(373\) 4.76271i 0.246604i 0.992369 + 0.123302i \(0.0393483\pi\)
−0.992369 + 0.123302i \(0.960652\pi\)
\(374\) −1.19269 −0.0616723
\(375\) 0 0
\(376\) −1.99330 −0.102796
\(377\) 7.78986i 0.401198i
\(378\) 1.79523i 0.0923368i
\(379\) −14.3773 −0.738514 −0.369257 0.929327i \(-0.620388\pi\)
−0.369257 + 0.929327i \(0.620388\pi\)
\(380\) 0 0
\(381\) 14.3308 0.734190
\(382\) − 1.46309i − 0.0748583i
\(383\) 30.6708i 1.56721i 0.621261 + 0.783603i \(0.286621\pi\)
−0.621261 + 0.783603i \(0.713379\pi\)
\(384\) 5.86353 0.299222
\(385\) 0 0
\(386\) 1.09544 0.0557565
\(387\) − 17.2470i − 0.876713i
\(388\) 3.01507i 0.153067i
\(389\) 12.9215 0.655148 0.327574 0.944825i \(-0.393769\pi\)
0.327574 + 0.944825i \(0.393769\pi\)
\(390\) 0 0
\(391\) 22.4916 1.13745
\(392\) − 4.02475i − 0.203281i
\(393\) 5.96854i 0.301073i
\(394\) 4.06829 0.204958
\(395\) 0 0
\(396\) −4.16719 −0.209409
\(397\) − 29.8471i − 1.49798i −0.662579 0.748992i \(-0.730538\pi\)
0.662579 0.748992i \(-0.269462\pi\)
\(398\) − 6.02954i − 0.302234i
\(399\) −1.35690 −0.0679298
\(400\) 0 0
\(401\) −28.7101 −1.43371 −0.716856 0.697221i \(-0.754420\pi\)
−0.716856 + 0.697221i \(0.754420\pi\)
\(402\) − 2.77240i − 0.138275i
\(403\) − 2.84415i − 0.141677i
\(404\) −32.1903 −1.60153
\(405\) 0 0
\(406\) −2.09352 −0.103900
\(407\) − 5.74094i − 0.284568i
\(408\) 4.13169i 0.204549i
\(409\) 13.6203 0.673479 0.336739 0.941598i \(-0.390676\pi\)
0.336739 + 0.941598i \(0.390676\pi\)
\(410\) 0 0
\(411\) −4.56033 −0.224945
\(412\) 9.39539i 0.462878i
\(413\) − 16.7138i − 0.822432i
\(414\) −2.47219 −0.121501
\(415\) 0 0
\(416\) −4.42253 −0.216833
\(417\) − 2.89977i − 0.142002i
\(418\) 0.225209i 0.0110153i
\(419\) 2.13946 0.104519 0.0522596 0.998634i \(-0.483358\pi\)
0.0522596 + 0.998634i \(0.483358\pi\)
\(420\) 0 0
\(421\) −15.0562 −0.733795 −0.366897 0.930261i \(-0.619580\pi\)
−0.366897 + 0.930261i \(0.619580\pi\)
\(422\) 1.07202i 0.0521849i
\(423\) − 4.82908i − 0.234798i
\(424\) 2.62837 0.127645
\(425\) 0 0
\(426\) −2.55065 −0.123579
\(427\) 0.918559i 0.0444522i
\(428\) − 8.66115i − 0.418653i
\(429\) −1.13706 −0.0548979
\(430\) 0 0
\(431\) 4.37435 0.210705 0.105353 0.994435i \(-0.466403\pi\)
0.105353 + 0.994435i \(0.466403\pi\)
\(432\) 15.6273i 0.751869i
\(433\) 33.1564i 1.59340i 0.604377 + 0.796698i \(0.293422\pi\)
−0.604377 + 0.796698i \(0.706578\pi\)
\(434\) 0.764365 0.0366907
\(435\) 0 0
\(436\) −36.5050 −1.74827
\(437\) − 4.24698i − 0.203161i
\(438\) 0.556023i 0.0265678i
\(439\) −32.2368 −1.53858 −0.769290 0.638900i \(-0.779390\pi\)
−0.769290 + 0.638900i \(0.779390\pi\)
\(440\) 0 0
\(441\) 9.75063 0.464316
\(442\) − 2.03385i − 0.0967405i
\(443\) − 21.7362i − 1.03272i −0.856373 0.516358i \(-0.827287\pi\)
0.856373 0.516358i \(-0.172713\pi\)
\(444\) −9.78986 −0.464606
\(445\) 0 0
\(446\) −6.48022 −0.306847
\(447\) − 2.63879i − 0.124811i
\(448\) 11.1217i 0.525450i
\(449\) 24.9584 1.17786 0.588929 0.808185i \(-0.299550\pi\)
0.588929 + 0.808185i \(0.299550\pi\)
\(450\) 0 0
\(451\) −3.81210 −0.179505
\(452\) − 38.8514i − 1.82742i
\(453\) − 8.19029i − 0.384814i
\(454\) 3.62697 0.170222
\(455\) 0 0
\(456\) 0.780167 0.0365347
\(457\) 13.1347i 0.614414i 0.951643 + 0.307207i \(0.0993944\pi\)
−0.951643 + 0.307207i \(0.900606\pi\)
\(458\) − 5.34481i − 0.249747i
\(459\) −22.7506 −1.06191
\(460\) 0 0
\(461\) −35.3726 −1.64746 −0.823732 0.566979i \(-0.808112\pi\)
−0.823732 + 0.566979i \(0.808112\pi\)
\(462\) − 0.305586i − 0.0142171i
\(463\) − 14.6963i − 0.682997i −0.939882 0.341498i \(-0.889066\pi\)
0.939882 0.341498i \(-0.110934\pi\)
\(464\) −18.2239 −0.846022
\(465\) 0 0
\(466\) −6.69766 −0.310263
\(467\) − 33.2121i − 1.53687i −0.639927 0.768435i \(-0.721036\pi\)
0.639927 0.768435i \(-0.278964\pi\)
\(468\) − 7.10620i − 0.328484i
\(469\) 23.6843 1.09364
\(470\) 0 0
\(471\) −11.5036 −0.530060
\(472\) 9.60984i 0.442329i
\(473\) 6.67264i 0.306809i
\(474\) −0.315748 −0.0145028
\(475\) 0 0
\(476\) −17.3749 −0.796379
\(477\) 6.36765i 0.291555i
\(478\) − 2.84787i − 0.130259i
\(479\) −24.6219 −1.12500 −0.562502 0.826796i \(-0.690161\pi\)
−0.562502 + 0.826796i \(0.690161\pi\)
\(480\) 0 0
\(481\) 9.78986 0.446379
\(482\) − 2.91915i − 0.132964i
\(483\) 5.76271i 0.262212i
\(484\) −19.7168 −0.896217
\(485\) 0 0
\(486\) 3.90110 0.176958
\(487\) − 29.5646i − 1.33970i −0.742496 0.669851i \(-0.766358\pi\)
0.742496 0.669851i \(-0.233642\pi\)
\(488\) − 0.528139i − 0.0239077i
\(489\) −15.6625 −0.708282
\(490\) 0 0
\(491\) 36.2978 1.63810 0.819049 0.573724i \(-0.194502\pi\)
0.819049 + 0.573724i \(0.194502\pi\)
\(492\) 6.50066i 0.293073i
\(493\) − 26.5308i − 1.19489i
\(494\) −0.384043 −0.0172789
\(495\) 0 0
\(496\) 6.65371 0.298760
\(497\) − 21.7899i − 0.977409i
\(498\) − 2.42327i − 0.108589i
\(499\) −7.84415 −0.351152 −0.175576 0.984466i \(-0.556179\pi\)
−0.175576 + 0.984466i \(0.556179\pi\)
\(500\) 0 0
\(501\) −9.52888 −0.425719
\(502\) 2.38703i 0.106538i
\(503\) 20.4166i 0.910330i 0.890407 + 0.455165i \(0.150420\pi\)
−0.890407 + 0.455165i \(0.849580\pi\)
\(504\) 3.87966 0.172814
\(505\) 0 0
\(506\) 0.956459 0.0425198
\(507\) 8.48619i 0.376885i
\(508\) − 34.6504i − 1.53736i
\(509\) 14.7530 0.653916 0.326958 0.945039i \(-0.393976\pi\)
0.326958 + 0.945039i \(0.393976\pi\)
\(510\) 0 0
\(511\) −4.75004 −0.210129
\(512\) − 17.4242i − 0.770048i
\(513\) 4.29590i 0.189668i
\(514\) 3.50365 0.154539
\(515\) 0 0
\(516\) 11.3787 0.500918
\(517\) 1.86831i 0.0821683i
\(518\) 2.63102i 0.115600i
\(519\) 12.4077 0.544639
\(520\) 0 0
\(521\) 37.1487 1.62751 0.813756 0.581206i \(-0.197419\pi\)
0.813756 + 0.581206i \(0.197419\pi\)
\(522\) 2.91617i 0.127637i
\(523\) 16.3623i 0.715472i 0.933823 + 0.357736i \(0.116451\pi\)
−0.933823 + 0.357736i \(0.883549\pi\)
\(524\) 14.4313 0.630434
\(525\) 0 0
\(526\) 5.38271 0.234698
\(527\) 9.68664i 0.421957i
\(528\) − 2.66009i − 0.115765i
\(529\) 4.96316 0.215790
\(530\) 0 0
\(531\) −23.2814 −1.01033
\(532\) 3.28083i 0.142242i
\(533\) − 6.50066i − 0.281575i
\(534\) −0.577793 −0.0250036
\(535\) 0 0
\(536\) −13.6176 −0.588191
\(537\) − 1.73556i − 0.0748950i
\(538\) − 6.10693i − 0.263289i
\(539\) −3.77240 −0.162489
\(540\) 0 0
\(541\) −2.08947 −0.0898335 −0.0449168 0.998991i \(-0.514302\pi\)
−0.0449168 + 0.998991i \(0.514302\pi\)
\(542\) 3.26742i 0.140348i
\(543\) 13.5506i 0.581514i
\(544\) 15.0623 0.645792
\(545\) 0 0
\(546\) 0.521106 0.0223013
\(547\) − 1.86054i − 0.0795511i −0.999209 0.0397756i \(-0.987336\pi\)
0.999209 0.0397756i \(-0.0126643\pi\)
\(548\) 11.0264i 0.471025i
\(549\) 1.27950 0.0546079
\(550\) 0 0
\(551\) −5.00969 −0.213420
\(552\) − 3.31336i − 0.141026i
\(553\) − 2.69740i − 0.114705i
\(554\) −0.138391 −0.00587968
\(555\) 0 0
\(556\) −7.01134 −0.297347
\(557\) 9.80061i 0.415265i 0.978207 + 0.207633i \(0.0665758\pi\)
−0.978207 + 0.207633i \(0.933424\pi\)
\(558\) − 1.06472i − 0.0450732i
\(559\) −11.3787 −0.481266
\(560\) 0 0
\(561\) 3.87263 0.163502
\(562\) − 6.81759i − 0.287583i
\(563\) 38.0170i 1.60222i 0.598514 + 0.801112i \(0.295758\pi\)
−0.598514 + 0.801112i \(0.704242\pi\)
\(564\) 3.18598 0.134154
\(565\) 0 0
\(566\) 3.94198 0.165694
\(567\) 6.13467i 0.257632i
\(568\) 12.5284i 0.525680i
\(569\) 7.32975 0.307279 0.153640 0.988127i \(-0.450901\pi\)
0.153640 + 0.988127i \(0.450901\pi\)
\(570\) 0 0
\(571\) 37.8775 1.58513 0.792563 0.609791i \(-0.208746\pi\)
0.792563 + 0.609791i \(0.208746\pi\)
\(572\) 2.74930i 0.114954i
\(573\) 4.75063i 0.198460i
\(574\) 1.74705 0.0729206
\(575\) 0 0
\(576\) 15.4919 0.645496
\(577\) − 28.6993i − 1.19477i −0.801955 0.597384i \(-0.796207\pi\)
0.801955 0.597384i \(-0.203793\pi\)
\(578\) 2.72827i 0.113481i
\(579\) −3.55688 −0.147819
\(580\) 0 0
\(581\) 20.7017 0.858852
\(582\) 0.307979i 0.0127661i
\(583\) − 2.46357i − 0.102030i
\(584\) 2.73110 0.113014
\(585\) 0 0
\(586\) −6.25667 −0.258461
\(587\) − 3.72348i − 0.153684i −0.997043 0.0768422i \(-0.975516\pi\)
0.997043 0.0768422i \(-0.0244838\pi\)
\(588\) 6.43296i 0.265291i
\(589\) 1.82908 0.0753661
\(590\) 0 0
\(591\) −13.2097 −0.543373
\(592\) 22.9028i 0.941297i
\(593\) 27.7399i 1.13914i 0.821943 + 0.569570i \(0.192890\pi\)
−0.821943 + 0.569570i \(0.807110\pi\)
\(594\) −0.967476 −0.0396960
\(595\) 0 0
\(596\) −6.38032 −0.261348
\(597\) 19.5778i 0.801266i
\(598\) 1.63102i 0.0666975i
\(599\) 23.5579 0.962551 0.481276 0.876569i \(-0.340174\pi\)
0.481276 + 0.876569i \(0.340174\pi\)
\(600\) 0 0
\(601\) −7.36898 −0.300587 −0.150293 0.988641i \(-0.548022\pi\)
−0.150293 + 0.988641i \(0.548022\pi\)
\(602\) − 3.05802i − 0.124635i
\(603\) − 32.9909i − 1.34349i
\(604\) −19.8033 −0.805783
\(605\) 0 0
\(606\) −3.28813 −0.133571
\(607\) 28.4198i 1.15352i 0.816912 + 0.576762i \(0.195684\pi\)
−0.816912 + 0.576762i \(0.804316\pi\)
\(608\) − 2.84415i − 0.115346i
\(609\) 6.79763 0.275454
\(610\) 0 0
\(611\) −3.18598 −0.128891
\(612\) 24.2024i 0.978323i
\(613\) − 6.48129i − 0.261777i −0.991397 0.130888i \(-0.958217\pi\)
0.991397 0.130888i \(-0.0417829\pi\)
\(614\) −2.85443 −0.115195
\(615\) 0 0
\(616\) −1.50099 −0.0604767
\(617\) − 24.4650i − 0.984924i −0.870334 0.492462i \(-0.836097\pi\)
0.870334 0.492462i \(-0.163903\pi\)
\(618\) 0.959706i 0.0386051i
\(619\) 22.2457 0.894128 0.447064 0.894502i \(-0.352470\pi\)
0.447064 + 0.894502i \(0.352470\pi\)
\(620\) 0 0
\(621\) 18.2446 0.732130
\(622\) 4.53020i 0.181645i
\(623\) − 4.93602i − 0.197757i
\(624\) 4.53617 0.181592
\(625\) 0 0
\(626\) 4.67084 0.186684
\(627\) − 0.731250i − 0.0292033i
\(628\) 27.8146i 1.10992i
\(629\) −33.3424 −1.32945
\(630\) 0 0
\(631\) −15.5888 −0.620581 −0.310290 0.950642i \(-0.600426\pi\)
−0.310290 + 0.950642i \(0.600426\pi\)
\(632\) 1.55091i 0.0616919i
\(633\) − 3.48081i − 0.138350i
\(634\) 5.00836 0.198907
\(635\) 0 0
\(636\) −4.20105 −0.166582
\(637\) − 6.43296i − 0.254883i
\(638\) − 1.12823i − 0.0446670i
\(639\) −30.3521 −1.20071
\(640\) 0 0
\(641\) 8.17496 0.322892 0.161446 0.986882i \(-0.448384\pi\)
0.161446 + 0.986882i \(0.448384\pi\)
\(642\) − 0.884707i − 0.0349166i
\(643\) − 11.1836i − 0.441038i −0.975383 0.220519i \(-0.929225\pi\)
0.975383 0.220519i \(-0.0707750\pi\)
\(644\) 13.9336 0.549062
\(645\) 0 0
\(646\) 1.30798 0.0514617
\(647\) 16.2121i 0.637362i 0.947862 + 0.318681i \(0.103240\pi\)
−0.947862 + 0.318681i \(0.896760\pi\)
\(648\) − 3.52722i − 0.138562i
\(649\) 9.00730 0.353567
\(650\) 0 0
\(651\) −2.48188 −0.0972725
\(652\) 37.8702i 1.48311i
\(653\) − 24.7952i − 0.970312i −0.874427 0.485156i \(-0.838763\pi\)
0.874427 0.485156i \(-0.161237\pi\)
\(654\) −3.72886 −0.145810
\(655\) 0 0
\(656\) 15.2079 0.593769
\(657\) 6.61655i 0.258136i
\(658\) − 0.856232i − 0.0333794i
\(659\) 20.2868 0.790262 0.395131 0.918625i \(-0.370699\pi\)
0.395131 + 0.918625i \(0.370699\pi\)
\(660\) 0 0
\(661\) 20.4222 0.794332 0.397166 0.917747i \(-0.369994\pi\)
0.397166 + 0.917747i \(0.369994\pi\)
\(662\) 1.17928i 0.0458339i
\(663\) 6.60388i 0.256473i
\(664\) −11.9028 −0.461917
\(665\) 0 0
\(666\) 3.66487 0.142011
\(667\) 21.2760i 0.823812i
\(668\) 23.0398i 0.891437i
\(669\) 21.0411 0.813498
\(670\) 0 0
\(671\) −0.495024 −0.0191102
\(672\) 3.85922i 0.148872i
\(673\) − 28.7254i − 1.10728i −0.832755 0.553641i \(-0.813238\pi\)
0.832755 0.553641i \(-0.186762\pi\)
\(674\) 6.02549 0.232093
\(675\) 0 0
\(676\) 20.5187 0.789181
\(677\) 44.0062i 1.69130i 0.533740 + 0.845648i \(0.320786\pi\)
−0.533740 + 0.845648i \(0.679214\pi\)
\(678\) − 3.96854i − 0.152411i
\(679\) −2.63102 −0.100969
\(680\) 0 0
\(681\) −11.7767 −0.451284
\(682\) 0.411927i 0.0157735i
\(683\) − 8.48427i − 0.324642i −0.986738 0.162321i \(-0.948102\pi\)
0.986738 0.162321i \(-0.0518979\pi\)
\(684\) 4.57002 0.174739
\(685\) 0 0
\(686\) 4.65412 0.177695
\(687\) 17.3545i 0.662116i
\(688\) − 26.6197i − 1.01487i
\(689\) 4.20105 0.160047
\(690\) 0 0
\(691\) 20.2586 0.770673 0.385336 0.922776i \(-0.374085\pi\)
0.385336 + 0.922776i \(0.374085\pi\)
\(692\) − 30.0006i − 1.14045i
\(693\) − 3.63640i − 0.138135i
\(694\) −2.46921 −0.0937297
\(695\) 0 0
\(696\) −3.90840 −0.148147
\(697\) 22.1400i 0.838614i
\(698\) 5.42327i 0.205274i
\(699\) 21.7472 0.822553
\(700\) 0 0
\(701\) −23.4101 −0.884188 −0.442094 0.896969i \(-0.645764\pi\)
−0.442094 + 0.896969i \(0.645764\pi\)
\(702\) − 1.64981i − 0.0622680i
\(703\) 6.29590i 0.237454i
\(704\) −5.99362 −0.225893
\(705\) 0 0
\(706\) 9.10826 0.342794
\(707\) − 28.0901i − 1.05644i
\(708\) − 15.3599i − 0.577260i
\(709\) −30.3472 −1.13971 −0.569857 0.821744i \(-0.693001\pi\)
−0.569857 + 0.821744i \(0.693001\pi\)
\(710\) 0 0
\(711\) −3.75733 −0.140911
\(712\) 2.83804i 0.106360i
\(713\) − 7.76809i − 0.290917i
\(714\) −1.77479 −0.0664199
\(715\) 0 0
\(716\) −4.19641 −0.156827
\(717\) 9.24698i 0.345335i
\(718\) 4.07979i 0.152256i
\(719\) 38.3230 1.42921 0.714604 0.699529i \(-0.246607\pi\)
0.714604 + 0.699529i \(0.246607\pi\)
\(720\) 0 0
\(721\) −8.19865 −0.305334
\(722\) − 0.246980i − 0.00919163i
\(723\) 9.47842i 0.352506i
\(724\) 32.7640 1.21767
\(725\) 0 0
\(726\) −2.01400 −0.0747466
\(727\) 2.69069i 0.0997923i 0.998754 + 0.0498961i \(0.0158890\pi\)
−0.998754 + 0.0498961i \(0.984111\pi\)
\(728\) − 2.55960i − 0.0948650i
\(729\) −1.78986 −0.0662910
\(730\) 0 0
\(731\) 38.7536 1.43335
\(732\) 0.844150i 0.0312007i
\(733\) − 18.9651i − 0.700491i −0.936658 0.350246i \(-0.886098\pi\)
0.936658 0.350246i \(-0.113902\pi\)
\(734\) −1.68724 −0.0622770
\(735\) 0 0
\(736\) −12.0790 −0.445240
\(737\) 12.7638i 0.470160i
\(738\) − 2.43355i − 0.0895803i
\(739\) −29.8278 −1.09723 −0.548616 0.836075i \(-0.684845\pi\)
−0.548616 + 0.836075i \(0.684845\pi\)
\(740\) 0 0
\(741\) 1.24698 0.0458089
\(742\) 1.12903i 0.0414480i
\(743\) − 8.78448i − 0.322271i −0.986932 0.161136i \(-0.948484\pi\)
0.986932 0.161136i \(-0.0515157\pi\)
\(744\) 1.42699 0.0523161
\(745\) 0 0
\(746\) 1.17629 0.0430671
\(747\) − 28.8364i − 1.05507i
\(748\) − 9.36360i − 0.342367i
\(749\) 7.55794 0.276161
\(750\) 0 0
\(751\) −18.1142 −0.660998 −0.330499 0.943806i \(-0.607217\pi\)
−0.330499 + 0.943806i \(0.607217\pi\)
\(752\) − 7.45340i − 0.271798i
\(753\) − 7.75063i − 0.282449i
\(754\) 1.92394 0.0700656
\(755\) 0 0
\(756\) −14.0941 −0.512598
\(757\) − 0.222816i − 0.00809840i −0.999992 0.00404920i \(-0.998711\pi\)
0.999992 0.00404920i \(-0.00128890\pi\)
\(758\) 3.55091i 0.128975i
\(759\) −3.10560 −0.112726
\(760\) 0 0
\(761\) −6.07798 −0.220327 −0.110163 0.993913i \(-0.535137\pi\)
−0.110163 + 0.993913i \(0.535137\pi\)
\(762\) − 3.53942i − 0.128220i
\(763\) − 31.8552i − 1.15323i
\(764\) 11.4865 0.415568
\(765\) 0 0
\(766\) 7.57507 0.273698
\(767\) 15.3599i 0.554613i
\(768\) 9.09411i 0.328156i
\(769\) 14.6267 0.527453 0.263726 0.964598i \(-0.415048\pi\)
0.263726 + 0.964598i \(0.415048\pi\)
\(770\) 0 0
\(771\) −11.3763 −0.409706
\(772\) 8.60015i 0.309526i
\(773\) 4.05728i 0.145930i 0.997334 + 0.0729651i \(0.0232462\pi\)
−0.997334 + 0.0729651i \(0.976754\pi\)
\(774\) −4.25965 −0.153110
\(775\) 0 0
\(776\) 1.51275 0.0543044
\(777\) − 8.54288i − 0.306474i
\(778\) − 3.19136i − 0.114416i
\(779\) 4.18060 0.149786
\(780\) 0 0
\(781\) 11.7429 0.420192
\(782\) − 5.55496i − 0.198645i
\(783\) − 21.5211i − 0.769102i
\(784\) 15.0495 0.537482
\(785\) 0 0
\(786\) 1.47411 0.0525797
\(787\) 19.5657i 0.697442i 0.937227 + 0.348721i \(0.113384\pi\)
−0.937227 + 0.348721i \(0.886616\pi\)
\(788\) 31.9396i 1.13780i
\(789\) −17.4776 −0.622218
\(790\) 0 0
\(791\) 33.9028 1.20544
\(792\) 2.09080i 0.0742934i
\(793\) − 0.844150i − 0.0299767i
\(794\) −7.37163 −0.261609
\(795\) 0 0
\(796\) 47.3370 1.67782
\(797\) 38.9051i 1.37809i 0.724718 + 0.689046i \(0.241970\pi\)
−0.724718 + 0.689046i \(0.758030\pi\)
\(798\) 0.335126i 0.0118633i
\(799\) 10.8509 0.383876
\(800\) 0 0
\(801\) −6.87561 −0.242938
\(802\) 7.09080i 0.250385i
\(803\) − 2.55986i − 0.0903355i
\(804\) 21.7657 0.767617
\(805\) 0 0
\(806\) −0.702447 −0.0247426
\(807\) 19.8291i 0.698017i
\(808\) 16.1508i 0.568183i
\(809\) 22.5730 0.793625 0.396812 0.917900i \(-0.370116\pi\)
0.396812 + 0.917900i \(0.370116\pi\)
\(810\) 0 0
\(811\) −46.3605 −1.62794 −0.813968 0.580909i \(-0.802697\pi\)
−0.813968 + 0.580909i \(0.802697\pi\)
\(812\) − 16.4359i − 0.576789i
\(813\) − 10.6093i − 0.372083i
\(814\) −1.41789 −0.0496972
\(815\) 0 0
\(816\) −15.4494 −0.540836
\(817\) − 7.31767i − 0.256013i
\(818\) − 3.36393i − 0.117617i
\(819\) 6.20105 0.216682
\(820\) 0 0
\(821\) −17.8194 −0.621901 −0.310951 0.950426i \(-0.600647\pi\)
−0.310951 + 0.950426i \(0.600647\pi\)
\(822\) 1.12631i 0.0392846i
\(823\) 32.5394i 1.13425i 0.823631 + 0.567126i \(0.191945\pi\)
−0.823631 + 0.567126i \(0.808055\pi\)
\(824\) 4.71394 0.164218
\(825\) 0 0
\(826\) −4.12797 −0.143630
\(827\) 39.8256i 1.38487i 0.721479 + 0.692436i \(0.243463\pi\)
−0.721479 + 0.692436i \(0.756537\pi\)
\(828\) − 19.4088i − 0.674502i
\(829\) −38.7525 −1.34593 −0.672966 0.739674i \(-0.734980\pi\)
−0.672966 + 0.739674i \(0.734980\pi\)
\(830\) 0 0
\(831\) 0.449354 0.0155879
\(832\) − 10.2208i − 0.354341i
\(833\) 21.9095i 0.759118i
\(834\) −0.716185 −0.0247994
\(835\) 0 0
\(836\) −1.76809 −0.0611505
\(837\) 7.85756i 0.271597i
\(838\) − 0.528402i − 0.0182533i
\(839\) −2.76749 −0.0955445 −0.0477723 0.998858i \(-0.515212\pi\)
−0.0477723 + 0.998858i \(0.515212\pi\)
\(840\) 0 0
\(841\) −3.90302 −0.134587
\(842\) 3.71858i 0.128151i
\(843\) 22.1366i 0.762425i
\(844\) −8.41624 −0.289699
\(845\) 0 0
\(846\) −1.19269 −0.0410054
\(847\) − 17.2054i − 0.591183i
\(848\) 9.82808i 0.337498i
\(849\) −12.7995 −0.439279
\(850\) 0 0
\(851\) 26.7385 0.916586
\(852\) − 20.0248i − 0.686037i
\(853\) − 24.1390i − 0.826503i −0.910617 0.413252i \(-0.864393\pi\)
0.910617 0.413252i \(-0.135607\pi\)
\(854\) 0.226865 0.00776317
\(855\) 0 0
\(856\) −4.34555 −0.148528
\(857\) − 3.16229i − 0.108022i −0.998540 0.0540109i \(-0.982799\pi\)
0.998540 0.0540109i \(-0.0172006\pi\)
\(858\) 0.280831i 0.00958743i
\(859\) −4.86426 −0.165967 −0.0829833 0.996551i \(-0.526445\pi\)
−0.0829833 + 0.996551i \(0.526445\pi\)
\(860\) 0 0
\(861\) −5.67264 −0.193323
\(862\) − 1.08038i − 0.0367978i
\(863\) − 4.54958i − 0.154870i −0.996997 0.0774348i \(-0.975327\pi\)
0.996997 0.0774348i \(-0.0246730\pi\)
\(864\) 12.2182 0.415671
\(865\) 0 0
\(866\) 8.18896 0.278272
\(867\) − 8.85862i − 0.300855i
\(868\) 6.00092i 0.203684i
\(869\) 1.45367 0.0493122
\(870\) 0 0
\(871\) −21.7657 −0.737502
\(872\) 18.3156i 0.620245i
\(873\) 3.66487i 0.124037i
\(874\) −1.04892 −0.0354802
\(875\) 0 0
\(876\) −4.36526 −0.147488
\(877\) − 18.6595i − 0.630086i −0.949077 0.315043i \(-0.897981\pi\)
0.949077 0.315043i \(-0.102019\pi\)
\(878\) 7.96184i 0.268699i
\(879\) 20.3153 0.685217
\(880\) 0 0
\(881\) 19.4306 0.654632 0.327316 0.944915i \(-0.393856\pi\)
0.327316 + 0.944915i \(0.393856\pi\)
\(882\) − 2.40821i − 0.0810885i
\(883\) − 25.6437i − 0.862979i −0.902118 0.431490i \(-0.857988\pi\)
0.902118 0.431490i \(-0.142012\pi\)
\(884\) 15.9675 0.537044
\(885\) 0 0
\(886\) −5.36839 −0.180354
\(887\) − 31.0062i − 1.04109i −0.853835 0.520544i \(-0.825729\pi\)
0.853835 0.520544i \(-0.174271\pi\)
\(888\) 4.91185i 0.164831i
\(889\) 30.2368 1.01411
\(890\) 0 0
\(891\) −3.30606 −0.110757
\(892\) − 50.8753i − 1.70343i
\(893\) − 2.04892i − 0.0685644i
\(894\) −0.651728 −0.0217970
\(895\) 0 0
\(896\) 12.3716 0.413305
\(897\) − 5.29590i − 0.176825i
\(898\) − 6.16421i − 0.205702i
\(899\) −9.16315 −0.305608
\(900\) 0 0
\(901\) −14.3080 −0.476668
\(902\) 0.941511i 0.0313489i
\(903\) 9.92931i 0.330427i
\(904\) −19.4929 −0.648324
\(905\) 0 0
\(906\) −2.02284 −0.0672042
\(907\) − 17.7676i − 0.589964i −0.955503 0.294982i \(-0.904686\pi\)
0.955503 0.294982i \(-0.0953136\pi\)
\(908\) 28.4748i 0.944971i
\(909\) −39.1280 −1.29779
\(910\) 0 0
\(911\) 41.7313 1.38262 0.691309 0.722559i \(-0.257034\pi\)
0.691309 + 0.722559i \(0.257034\pi\)
\(912\) 2.91723i 0.0965992i
\(913\) 11.1564i 0.369224i
\(914\) 3.24400 0.107302
\(915\) 0 0
\(916\) 41.9614 1.38644
\(917\) 12.5931i 0.415862i
\(918\) 5.61894i 0.185453i
\(919\) −30.4088 −1.00309 −0.501547 0.865130i \(-0.667236\pi\)
−0.501547 + 0.865130i \(0.667236\pi\)
\(920\) 0 0
\(921\) 9.26828 0.305400
\(922\) 8.73630i 0.287715i
\(923\) 20.0248i 0.659123i
\(924\) 2.39911 0.0789249
\(925\) 0 0
\(926\) −3.62969 −0.119279
\(927\) 11.4203i 0.375091i
\(928\) 14.2483i 0.467724i
\(929\) 28.8219 0.945616 0.472808 0.881165i \(-0.343240\pi\)
0.472808 + 0.881165i \(0.343240\pi\)
\(930\) 0 0
\(931\) 4.13706 0.135587
\(932\) − 52.5824i − 1.72239i
\(933\) − 14.7095i − 0.481567i
\(934\) −8.20270 −0.268401
\(935\) 0 0
\(936\) −3.56538 −0.116538
\(937\) − 50.4601i − 1.64846i −0.566255 0.824230i \(-0.691608\pi\)
0.566255 0.824230i \(-0.308392\pi\)
\(938\) − 5.84953i − 0.190994i
\(939\) −15.1661 −0.494928
\(940\) 0 0
\(941\) 1.10082 0.0358857 0.0179428 0.999839i \(-0.494288\pi\)
0.0179428 + 0.999839i \(0.494288\pi\)
\(942\) 2.84117i 0.0925702i
\(943\) − 17.7549i − 0.578180i
\(944\) −35.9335 −1.16954
\(945\) 0 0
\(946\) 1.64801 0.0535813
\(947\) − 13.9353i − 0.452836i −0.974030 0.226418i \(-0.927299\pi\)
0.974030 0.226418i \(-0.0727015\pi\)
\(948\) − 2.47889i − 0.0805107i
\(949\) 4.36526 0.141702
\(950\) 0 0
\(951\) −16.2620 −0.527333
\(952\) 8.71751i 0.282536i
\(953\) 41.3400i 1.33913i 0.742751 + 0.669567i \(0.233520\pi\)
−0.742751 + 0.669567i \(0.766480\pi\)
\(954\) 1.57268 0.0509174
\(955\) 0 0
\(956\) 22.3582 0.723117
\(957\) 3.66334i 0.118419i
\(958\) 6.08111i 0.196472i
\(959\) −9.62192 −0.310708
\(960\) 0 0
\(961\) −27.6544 −0.892079
\(962\) − 2.41789i − 0.0779561i
\(963\) − 10.5278i − 0.339254i
\(964\) 22.9178 0.738133
\(965\) 0 0
\(966\) 1.42327 0.0457930
\(967\) − 5.26875i − 0.169432i −0.996405 0.0847158i \(-0.973002\pi\)
0.996405 0.0847158i \(-0.0269982\pi\)
\(968\) 9.89248i 0.317956i
\(969\) −4.24698 −0.136433
\(970\) 0 0
\(971\) −5.15346 −0.165382 −0.0826911 0.996575i \(-0.526351\pi\)
−0.0826911 + 0.996575i \(0.526351\pi\)
\(972\) 30.6270i 0.982361i
\(973\) − 6.11828i − 0.196143i
\(974\) −7.30186 −0.233967
\(975\) 0 0
\(976\) 1.97484 0.0632130
\(977\) 4.77612i 0.152802i 0.997077 + 0.0764008i \(0.0243428\pi\)
−0.997077 + 0.0764008i \(0.975657\pi\)
\(978\) 3.86831i 0.123695i
\(979\) 2.66009 0.0850168
\(980\) 0 0
\(981\) −44.3726 −1.41671
\(982\) − 8.96482i − 0.286079i
\(983\) 28.9758i 0.924186i 0.886832 + 0.462093i \(0.152901\pi\)
−0.886832 + 0.462093i \(0.847099\pi\)
\(984\) 3.26157 0.103975
\(985\) 0 0
\(986\) −6.55257 −0.208676
\(987\) 2.78017i 0.0884937i
\(988\) − 3.01507i − 0.0959220i
\(989\) −31.0780 −0.988222
\(990\) 0 0
\(991\) −38.5042 −1.22313 −0.611564 0.791195i \(-0.709459\pi\)
−0.611564 + 0.791195i \(0.709459\pi\)
\(992\) − 5.20219i − 0.165170i
\(993\) − 3.82908i − 0.121512i
\(994\) −5.38165 −0.170696
\(995\) 0 0
\(996\) 19.0248 0.602822
\(997\) 10.1491i 0.321427i 0.987001 + 0.160713i \(0.0513795\pi\)
−0.987001 + 0.160713i \(0.948621\pi\)
\(998\) 1.93735i 0.0613256i
\(999\) −27.0465 −0.855714
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 475.2.b.c.324.3 6
5.2 odd 4 475.2.a.d.1.3 3
5.3 odd 4 475.2.a.h.1.1 yes 3
5.4 even 2 inner 475.2.b.c.324.4 6
15.2 even 4 4275.2.a.bn.1.1 3
15.8 even 4 4275.2.a.z.1.3 3
20.3 even 4 7600.2.a.bn.1.3 3
20.7 even 4 7600.2.a.bw.1.1 3
95.18 even 4 9025.2.a.w.1.3 3
95.37 even 4 9025.2.a.be.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.d.1.3 3 5.2 odd 4
475.2.a.h.1.1 yes 3 5.3 odd 4
475.2.b.c.324.3 6 1.1 even 1 trivial
475.2.b.c.324.4 6 5.4 even 2 inner
4275.2.a.z.1.3 3 15.8 even 4
4275.2.a.bn.1.1 3 15.2 even 4
7600.2.a.bn.1.3 3 20.3 even 4
7600.2.a.bw.1.1 3 20.7 even 4
9025.2.a.w.1.3 3 95.18 even 4
9025.2.a.be.1.1 3 95.37 even 4